A067687 -id:A067687 - cp's OEIS frontend

A343234 Triangle T read by rows: lower triangular Riordan matrix of the Toeplitz type with first column A067687.

1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 12, 5, 2, 1, 1, 29, 12, 5, 2, 1, 1, 69, 29, 12, 5, 2, 1, 1, 165, 69, 29, 12, 5, 2, 1, 1, 393, 165, 69, 29, 12, 5, 2, 1, 1, 937, 393, 165, 69, 29, 12, 5, 2, 1, 1, 2233, 937, 393, 165, 69, 29, 12, 5, 2, 1, 1

Offset: 0

Wolfdieter Lang, Apr 16 2021

This infinite lower triangular Riordan matrix T is the so-called L-eigen-matrix of the infinite lower triangular Riordan matrix A027293 (but with offset 0 for rows and columns). Such eigentriangles have been considered by Paul Barry in the paper given as a link in A186020.
This means that E is the L-eigen-matrix of an infinite lower triangular matrix M if M*E = L*(E - I), with the unit matrix I and the matrix L with elements L(i, j) = delta_{i, j-1} (Kronecker's delta-symbol; first upper diagonal with 1's).
Therefore, this notion is analogous to calling sequence S an L-eigen-sequence of matrix M if M*vec(S) = L.vec(S) (or vec(S) is an eigensequence of M - L with eigenvalue 0), used by Bernstein and Sloane, see the links in A155002.
L*(E - I) is the E matrix after elimination of the main diagonal and then the first row, and starting with offset 0. Because for infinite lower triangular matrices L^{tr}.L = I (tr stands for transposed), this leads to M = L*(I - E^{-1}) or E = (I - L^{tr}*M)^{-1}.
Note that Gary W. Adamson uses a different notion: E is the eigentriangle of a triangle T if the columns of E are the columns j of T multiplied by the sequence elements B_j of B with o.g.f. x/(1 - x*G(x)), with the o.g.f. G(x) of column no. 1 of T.  Or E(i, j) = T(i, j)*B(j). In short: sequence B is the L-eigen-sequence of the infinite lower triangular matrix T (but with offset 1): T*vec(B) = L.vec(B). See, e.g., A143866.
Thanks to Gary W. Adamson for motivating my occupation with such eigentriangles and eigensequences.
The first column of the present triangle T is A067687, which is then shifted downwards (Riordan of Toeplitz type).

			The triangle T begins:
n \ m   0   1   2   3  4  5  6  7  8  9 ...
-----------------------------------------
0:      1
1:      1   1
2:      2   1   1
3:      5   2   1   1
4:     12   5   2   1  1
5:     29  12   5   2  1  1
6:     69  29  12   5  2  1  1
7:    165  69  29  12  5  2  1  1
8:    393 165  69  29 12  5  2  1  1
9:    937 393 165  69 29 12  5  2  1  1
...

Cf. A000041, A027293, A067687, A143866, A155002, A186020.

Matrix elements: T(n, m) = A067687(n-m), for n >= m >= 0, and 0 otherwise.
O.g.f. of row polynomials R(n,x) = Sum_{m=0..n} T(n, m)*x^m is
G(z, x) = 1/((1 - z*P(z))*(1 - x*z)), with the o.g.f. P of A000041 (number of partitions).
O.g.f. column m: G_m(x) = x^m/(1 - x*P(x)), for m >= 0.

A304969 Expansion of 1/(1 - Sum_{k>=1} q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, 1, 2, 5, 11, 25, 57, 129, 292, 662, 1500, 3398, 7699, 17443, 39519, 89536, 202855, 459593, 1041267, 2359122, 5344889, 12109524, 27435660, 62158961, 140828999, 319065932, 722884274, 1637785870, 3710611298, 8406859805, 19046805534, 43152950024, 97768473163

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Invert transform of A000009.
From Gus Wiseman, Jul 31 2022: (Start)
Also the number of ways to choose a multiset partition into distinct constant multisets of a multiset of length n that covers an initial interval of positive integers. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 11 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{1},{2,2}}    {{1,1},{2,2}}
                    {{2},{1,1}}    {{1},{2,2,2}}
                    {{1},{2},{3}}  {{2},{1,1,1}}
                                   {{1},{2},{1,1}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{3},{2,2}}
                                   {{2},{3},{1,1}}
                                   {{1},{2},{3},{4}}
The non-strict version is A055887.
The strongly normal non-strict version is A063834.
The strongly normal version is A270995.
(End)

			From _Gus Wiseman_, Jul 31 2022: (Start)
a(n) is the number of ways to choose a strict integer partition of each part of an integer composition of n. The a(1) = 1 through a(4) = 11 choices are:
  ((1))  ((2))     ((3))        ((4))
         ((1)(1))  ((21))       ((31))
                   ((1)(2))     ((1)(3))
                   ((2)(1))     ((2)(2))
                   ((1)(1)(1))  ((3)(1))
                                ((1)(21))
                                ((21)(1))
                                ((1)(1)(2))
                                ((1)(2)(1))
                                ((2)(1)(1))
                                ((1)(1)(1)(1))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2816
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Partition Function Q
Index entries for sequences related to partitions
Index entries for sequences related to compositions

Cf. A050342, A055887, A067687, A081362, A279785, A299106.
Row sums of A308680.
The unordered version is A089259, non-strict A001970 (row-sums of A061260).
For partitions instead of compositions we have A270995, non-strict A063834.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by sum and length.
Cf. A279784.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(j)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

nmax = 32; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - 1/QPochhammer[x, x^2]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

G.f.: 1/(1 - Sum_{k>=1} A000009(k)*x^k).
G.f.: 1/(2 - Product_{k>=1} (1 + x^k)).
G.f.: 1/(2 - Product_{k>=1} 1/(1 - x^(2*k-1))).
G.f.: 1/(2 - exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)))).
a(n) ~ c / r^n, where r = 0.441378990861652015438479635503868737167721352874... is the root of the equation QPochhammer[-1, r] = 4 and c = 0.4208931614610039677452560636348863586180784719323982664940444607322... - Vaclav Kotesovec, May 23 2018

A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 5, 0, 1, 5, 14, 22, 20, 7, 0, 1, 6, 20, 40, 51, 36, 11, 0, 1, 7, 27, 65, 105, 108, 65, 15, 0, 1, 8, 35, 98, 190, 252, 221, 110, 22, 0, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 0, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 0

Offset: 0

Alois P. Heinz, Sep 09 2008

A(n,k) is also the number of partitions of n into parts of k kinds.
In general, column k > 0 is asymptotic to k^((k+1)/4) * exp(Pi*sqrt(2*k*n/3)) / (2^((3*k+5)/4) * 3^((k+1)/4) * n^((k+3)/4)) * (1 - (Pi*k^(3/2)/(24*sqrt(6)) + sqrt(3)*(k+1)*(k+3)/(8*Pi*sqrt(2*k))) / sqrt(n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017
When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field with k elements that contain an upper-triangular matrix. - Geoffrey Critzer, Nov 11 2022

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   2,   3,   4,   5, ...
  0,   2,   5,   9,  14,  20, ...
  0,   3,  10,  22,  40,  65, ...
  0,   5,  20,  51, 105, 190, ...
  0,   7,  36, 108, 252, 506, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48. See Table 1.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Columns k=0-24 give: A000007, A000041, A000712, A000716, A023003, A023004, A023005, A023006, A023007, A023008, A023009, A023010, A005758, A023011, A023012, A023013, A023014, A023015, A023016, A023017, A023018, A023019, A023020, A023021, A006922.
Cf. A082556 (k=30), A082557 (k=32), A082558 (k=48), A082559 (k=64).
Rows n=0-4 give: A000012, A001477, A000096, A006503, A006504.
Main diagonal gives A008485.
Antidiagonal sums give A067687.
Cf. A060642, A122768, A246935, A255961, A261718.

# DedekindEta is defined in A000594.
A144064Column(k, len) = DedekindEta(len, -k)
for n in 0:8 A144064Column(n, 6) |> println end # Peter Luschny, Mar 10 2018

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->k)(n): seq(seq(A(n, d-n), n=0..d), d=0..14);

a[0, ] = 1; a[, 0] = 0; a[n_, k_] := SeriesCoefficient[ Product[1/(1 - x^j)^k, {j, 1, n}], {x, 0, n}]; Table[a[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d*p[d], {d, Divisors[j]} ]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[k&][n]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)

Mat(apply( {A144064_col(k,nMax=9)=Col(1/eta('x+O('x^nMax))^k,nMax)}, [0..9])) \\ M. F. Hasler, Aug 04 2024

G.f. of column k: Product_{j>=1} 1/(1-x^j)^k.

A(n,k) = Sum_{i=0..k} binomial(k,i) * A060642(n,k-i):

A027293 Triangular array given by rows: P(n,k) is the number of partitions of n that contain k as a part.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 3, 2, 1, 1, 7, 5, 3, 2, 1, 1, 11, 7, 5, 3, 2, 1, 1, 15, 11, 7, 5, 3, 2, 1, 1, 22, 15, 11, 7, 5, 3, 2, 1, 1, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 77

Offset: 1

Clark Kimberling

Triangle read by rows in which row n lists the first n partition numbers A000041 in decreasing order. - Omar E. Pol, Aug 06 2011
A027293 * an infinite lower triangular matrix with A010815 (1, -1, -1, 0, 0, 1, ...) as the main diagonal the rest zeros = triangle A145975 having row sums = [1, 0, 0, 0, ...]. These matrix operations are equivalent to the comment in A010815 stating "when convolved with the partition numbers = [1, 0, 0, 0, ...]. - Gary W. Adamson, Oct 25 2008
From Gary W. Adamson, Oct 26 2008: (Start)
Row sums = A000070: (1, 2, 4, 7, 12, 19, 30, 45, 67, ...);
(this triangle)^2 = triangle A146023. (End)
(1) It appears that P(n,k) is also the total number of occurrences of k in the last k sections of the set of partitions of n (cf. A182703). (2) It appears that P(n,k) is also the difference, between n and n-k, of the total number of occurrences of k in all their partitions (cf. A066633). - Omar E. Pol, Feb 07 2012
Sequence B is called a reverse reluctant sequence of sequence A, if B is a triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. The present sequence is the reverse reluctant sequence of (A000041(k-1)){k>=0}. - _Boris Putievskiy, Dec 14 2012

			The triangle P begins (with offsets 0 it is Pa):
n \ k  1  2  3  4  5  6  7  8  9 10 ...
1:     1
2:     1  1
3:     2  1  1
4:     3  2  1  1
5:     5  3  2  1  1
6:     7  5  3  2  1  1
7:    11  7  5  3  2  1  1
8:    15 11  7  5  3  2  1  1
9:    22 15 11  7  5  3  2  1  1
10:   30 22 15 11  7  5  3  2  1  1
... reformatted by _Wolfdieter Lang_, Apr 14 2021

Robert Price, Table of n, a(n) for n = 1..5050
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Every column of P is A000041.
Cf. A145975, A010815.
Cf. A000070, A146023.
Cf. A182700, A027293, A067687, A155002.
Cf. A343234 (L-eigen-matrix).

f[n_] := Block[{t = Flatten[Union /@ IntegerPartitions@n]}, Table[Count[t, i], {i, n}]]; Array[f, 13] // Flatten
t[n_, k_] := PartitionsP[n-k]; Table[t[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 24 2014 *)

P(n,k) = p(n-k) = A000041(n-k), n>=1, k>=1. - Omar E. Pol, Feb 15 2013
a(n) = A000041(m), where m = (t*t + 3*t + 4)/2 - n, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 14 2012
From Wolfdieter Lang, Apr 14 2021: (Start)
Pa(n, m) = P(n+1, m+1) = A000041(n-m), for n >= m >= 0, and 0 otherwise, gives the Riordan matrix Pa = (P(x), x), of Toeplitz type, with the o.g.f. P(x) of A000041. The o.g.f. of triangle Pa (the o.g.f. of the row polynomials RPa(n, x) = Sum_{m=0..n} Pa(n, m)*x^m) is G(z, x) = P(z)/(1 - x*z).
The (infinite) matrix Pa has the 'L-eigen-sequence' B = A067687, that is, Pa*vec(B) = L*vec(B), with the matrix L with elements L(i, j) = delta(i, j-1) (Kronecker's delta symbol). For such L-eigen-sequences see the Bernstein and Sloane links under A155002.
Thanks to Gary W. Adamson for motivating me to look at such matrices and sequences. (End)

A299106 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)).

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 41, 88, 189, 405, 869, 1864, 3998, 8575, 18392, 39448, 84610, 181475, 389235, 834848, 1790617, 3840591, 8237462, 17668057, 37895195, 81279216, 174331098, 373912708, 801983781, 1720128713, 3689404772, 7913191304, 16972547194, 36403436640

Offset: 0

Ilya Gutkovskiy, Feb 02 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000
N. J. A. Sloane, Transforms

Antidiagonal sums of A286335.

Cf. A000009, A067687, A299105, A299108.

nmax = 33; CoefficientList[Series[1/(1 - x Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[1/(1 - x/QPochhammer[x, x^2]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A000009(k-1)*a(n-k).
a(n) ~ c * d^n, where d = 2.14484226934608840026733598736202689102117985119507858808036465196716739... is the root of the equation QPochhammer(1/d, 1/d^2)*d = 1 and c = 0.4217892515709863296976217395517853732959704351198250451894928058439... = 2/(2+Derivative[0, 1][QPochhammer][-1, 1/d]/d^2). - Vaclav Kotesovec, Feb 03 2018, updated Mar 31 2018

A299105 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)).

Original entry on oeis.org

1, 1, 0, -2, -3, -1, 5, 10, 7, -9, -29, -30, 10, 77, 108, 22, -184, -351, -207, 372, 1041, 969, -516, -2835, -3655, -284, 6990, 12190, 5977, -14957, -37044, -30994, 24144, 103374, 122409, -7715, -262704, -420585, -162274, 589068, 1309674, 972747, -1057935, -3742955

Offset: 0

Ilya Gutkovskiy, Feb 02 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Transforms

Antidiagonal sums of A286354.
Cf. similar sequences: A067687, A299106, A299208, A302017, A318581, A318582, A331484.
Cf. A010815, A299108.

nmax = 43; CoefficientList[Series[1/(1 - x Product[1 - x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 43; CoefficientList[Series[1/(1 - x QPochhammer[x, x]), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^k)).

a(0) = 1; a(n) = Sum_{k=1..n} A010815(k-1)*a(n-k).

A299108 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)).

Original entry on oeis.org

1, 1, 3, 9, 27, 79, 231, 675, 1971, 5755, 16805, 49071, 143289, 418411, 1221781, 3567663, 10417761, 30420401, 88829145, 259385701, 757419669, 2211704625, 6458291945, 18858546645, 55067931981, 160801210705, 469547855419, 1371104033121, 4003694720243

Offset: 0

Ilya Gutkovskiy, Feb 02 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Antidiagonal sums of A288515.

Cf. A015128, A032803, A067687, A299105, A299106.

S:= series(1/(1-x/JacobiTheta4(0,x)),x,51):
seq(coeff(S,x,n),n=0..50); # Robert Israel, Feb 02 2018

nmax = 28; CoefficientList[Series[1/(1 - x Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[1/(1 - x/EllipticTheta[4, 0, x]), {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[1/(1 - x QPochhammer[-x, x]/QPochhammer[x, x]), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)).
G.f.: 1/(1 - x/theta_4(x)), where theta_4() is the Jacobi theta function.
a(0) = 1; a(n) = Sum_{k=1..n} A015128(k-1)*a(n-k).
a(n) ~ c * d^n, where d = 2.9200517419026569743994130834319365190407162724411912701937027582419975778... is the root of the equation EllipticTheta(4, 0, 1/d) * d = 1 and c = 0.372842695601022868809531452599286285949969156503576039087883242107... = 2*Log[r]*QPochhammer[r] / (2*QPochhammer[r] * (Log[1 - r] + Log[r] + QPolyGamma[1, r]) + r*Log[r] * (r * Derivative[0, 1][QPochhammer][-1, r] - 2*Derivative[0, 1][QPochhammer][r, r])), where r = 1/d. Equivalently, c = EllipticTheta[4, 0, r]^2 / (r *(EllipticTheta[4, 0, r] - r * Derivative[0, 0, 1][EllipticTheta][4, 0, r])). - Vaclav Kotesovec, Feb 03 2018, updated Mar 31 2018

A299208 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

Original entry on oeis.org

1, 1, 0, -1, -2, -1, 1, 3, 3, 1, -3, -6, -5, 1, 9, 12, 5, -9, -20, -18, 1, 26, 38, 21, -21, -61, -62, -9, 72, 120, 81, -44, -177, -205, -64, 186, 366, 293, -63, -496, -657, -304, 445, 1084, 1014, 33, -1341, -2053, -1238, 959, 3132, 3378, 770, -3474, -6260, -4619, 1656, 8809, 10929, 4306, -8520

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Antidiagonal sums of A286352.
Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A318582, A331484.
Cf. A081362, A299108, A299162, A299164, A299166, A299167, A299209, A299210, A299211, A299212.

nmax = 60; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

a(0) = 1; a(n) = Sum_{k=1..n} A081362(k-1)*a(n-k).

A299162 Expansion of 1/(1 - xProduct_{k>=1} 1/(1 - kx^k)).

Original entry on oeis.org

1, 1, 2, 6, 17, 49, 135, 380, 1051, 2925, 8119, 22548, 62574, 173767, 482360, 1339126, 3717700, 10321163, 28653557, 79548612, 220843925, 613110573, 1702128034, 4725475979, 13118945083, 36421037100, 101112695940, 280710759278, 779313926949, 2163544401343, 6006468273440

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2256
N. J. A. Sloane, Transforms

Antidiagonal sums of A297328.

Cf. A006906, A067687, A299105, A299106, A299108, A299164, A299166, A299167.

nmax = 30; CoefficientList[Series[1/(1 - x Product[1/(1 - k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 - k*x^k)).

a(0) = 1; a(n) = Sum_{k=1..n} A006906(k-1)*a(n-k).

A299164 Expansion of 1/(1 - xProduct_{k>=1} (1 + kx^k)).

Original entry on oeis.org

1, 1, 2, 5, 14, 35, 91, 233, 597, 1517, 3885, 9922, 25333, 64683, 165181, 421828, 1077277, 2750993, 7025168, 17940298, 45814165, 116996152, 298774246, 762982615, 1948434235, 4975732669, 12706571546, 32448880807, 82864981016, 211613009498, 540397935771, 1380018797044, 3524165721799

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Antidiagonal sums of A297321.

Cf. A022629, A067687, A299105, A299106, A299108, A299162, A299166, A299167.

nmax = 32; CoefficientList[Series[1/(1 - x Product[1 + k x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} (1 + k*x^k)).

a(0) = 1; a(n) = Sum_{k=1..n} A022629(k-1)*a(n-k).

cp's OEIS Frontend

A343234 Triangle T read by rows: lower triangular Riordan matrix of the Toeplitz type with first column A067687.

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A304969 Expansion of 1/(1 - Sum_{k>=1} q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

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A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

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Julia

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A027293 Triangular array given by rows: P(n,k) is the number of partitions of n that contain k as a part.

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A299106 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)).

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A299105 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)).

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A299108 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)).

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A299208 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

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A299162 Expansion of 1/(1 - xProduct_{k>=1} 1/(1 - kx^k)).

A299164 Expansion of 1/(1 - xProduct_{k>=1} (1 + kx^k)).